Physics:f/sci.+engrs.,ap Ed.
10th Edition
ISBN: 9781337553469
Author: Jewett, SERWAY
Publisher: Cengage
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Chapter 29, Problem 15P
You are part of a team working in a machine parts mechanic’s shop. An important customer has asked your company to provide springs with a very precise force constant k. To measure the spring constant, you fasten two of the springs between the ends of two very long wires of length L, separated by the unstretched length ℓ of the springs as shown in Figure P29.15. The specific attachment method that you use insulates the springs from the wires so that no current passes through the springs. You lay the apparatus flat on a table and then pass a current of magnitude I through the wires, in opposite directions. As a result the springs stretch by a distance d and come to equilibrium. You determine an expression for the spring constant in terms of L, I, ℓ, and d.
Figure P29.15
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
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[5 marks]
(b)…
Chapter 29 Solutions
Physics:f/sci.+engrs.,ap Ed.
Ch. 29.1 - Consider the magnetic field due to the current in...Ch. 29.2 - A loose spiral spring carrying no current is hung...Ch. 29.3 - Prob. 29.3QQCh. 29.3 - Prob. 29.4QQCh. 29.4 - Consider a solenoid that is very long compared...Ch. 29 - Calculate the magnitude of the magnetic field at a...Ch. 29 - You are working as an expert witness in a civil...Ch. 29 - In Niels Bohrs 1913 model of the hydrogen atom, an...Ch. 29 - Prob. 4PCh. 29 - Prob. 5P
Ch. 29 - Consider a flat, circular current loop of radius R...Ch. 29 - Prob. 7PCh. 29 - One long wire carries current 30.0 A to the left...Ch. 29 - Determine the magnetic field (in terms of I, a,...Ch. 29 - Prob. 10PCh. 29 - Two long, parallel wires carry currents of I1 =...Ch. 29 - Prob. 12PCh. 29 - Prob. 13PCh. 29 - Prob. 14PCh. 29 - You are part of a team working in a machine parts...Ch. 29 - Why is the following situation impossible? Two...Ch. 29 - Prob. 17PCh. 29 - Niobium metal becomes a superconductor when cooled...Ch. 29 - The magnetic coils of a tokamak fusion reactor are...Ch. 29 - A packed bundle of 100 long, straight, insulated...Ch. 29 - The magnetic field 40.0 cm away from a long,...Ch. 29 - Prob. 22PCh. 29 - A long solenoid that has 1 000 turns uniformly...Ch. 29 - A certain superconducting magnet in the form of a...Ch. 29 - Prob. 25PCh. 29 - You are given a certain volume of copper from...Ch. 29 - Prob. 27PCh. 29 - You are working for a company that creates special...Ch. 29 - A solenoid of radius r = 1.25 cm and length =...Ch. 29 - The magnetic moment of the Earth is approximately...Ch. 29 - A 30.0-turn solenoid of length 6.00 cm produces a...Ch. 29 - Why is the following situation impossible? The...Ch. 29 - Suppose you install a compass on the center of a...Ch. 29 - Prob. 34APCh. 29 - A nonconducting ring of radius 10.0 cm is...Ch. 29 - Prob. 36APCh. 29 - A very large parallel-plate capacitor has uniform...Ch. 29 - Two circular coils of radius R, each with N turns,...Ch. 29 - Prob. 39APCh. 29 - Two circular loops are parallel, coaxial, and...Ch. 29 - As seen in previous chapters, any object with...Ch. 29 - Review. Rail guns have been suggested for...Ch. 29 - Prob. 43APCh. 29 - An infinitely long, straight wire carrying a...Ch. 29 - Prob. 45CPCh. 29 - We have seen that a long solenoid produces a...Ch. 29 - A wire carrying a current I is bent into the shape...Ch. 29 - Prob. 48CPCh. 29 - Prob. 49CPCh. 29 - Prob. 50CPCh. 29 - The magnitude of the force on a magnetic dipole ...
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